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Some relations on the lattice of varieties of completely regular semigroups

Mario Petrich (2002)

Bollettino dell'Unione Matematica Italiana

On the lattice L ⁢ C ⁢ R of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations K l , K , K r , T l , T , T r , C and L . Here K is the kernel relation, T is the trace relation, T l and T r are the left and the right trace relations, respectively, K p = K ∩ T p for p ∈ l , r , C is the core relation and L is the local relation. We give an alternative definition for each of these relations P of the form U P V ⇔ U ∩ P ~ = V ∩ P ~ ( U , V ∈ L ( C R ) ) , for some subclasses P ~ of C ⁢ R ....

Spaces and equations

Walter Taylor (2000)

Fundamenta Mathematicae

It is proved, for various spaces A, such as a surface of genus 2, a figure-eight, or a sphere of dimension ≠ 1,3,7, and for any set Σ of equations, that Σ cannot be modeled by continuous operations on A unless Σ is undemanding (a form of triviality that is defined in the paper).

Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)

Apinant Anantpinitwatna, Tiang Poomsa-ard (2009)

Discussiones Mathematicae - General Algebra and Applications

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A ( G ) ̲ satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = M o d g Σ where Σ is a subset of T(X) × T(X). A graph variety V ' = M o d g Σ ' is called a biregular leftmost graph variety if Σ’ is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety...

Sturdy frames of type (2,2) algebras and their applications to semirings

X. Z. Zhao, Y. Q. Guo, K. P. Shum (2003)

Fundamenta Mathematicae

We introduce sturdy frames of type (2,2) algebras, which are a common generalization of sturdy semilattices of semigroups and of distributive lattices of rings in the theory of semirings. By using sturdy frames, we are able to characterize some semirings. In particular, some results on semirings recently obtained by Bandelt, Petrich and Ghosh can be extended and generalized.

Subalgebras and homomorphic images of algebras having the CEP and the WCIP

Andrzej Walendziak (2004)

Czechoslovak Mathematical Journal

In the present paper we consider algebras satisfying both the congruence extension property (briefly the CEP) and the weak congruence intersection property (WCIP for short). We prove that subalgebras of such algebras have these properties. We deduce that a lattice has the CEP and the WCIP if and only if it is a two-element chain. We also show that the class of all congruence modular algebras with the WCIP is closed under the formation of homomorphic images.

Subdirect decompositions of algebras from 2-clone extensions of varieties

J. Płonka (1998)

Colloquium Mathematicae

Let τ:F → ℕ be a type of algebras, where F is a set of fundamental operation symbols and ℕ is the set of nonnegative integers. We assume that |F|≥2 and 0 ∉ (F). For a term φ of type τ we denote by F(φ) the set of fundamental operation symbols from F occurring in φ. An identity φ ≉ ψ of type τ is called clone compatible if φ and ψ are the same variable or F(φ)=F(ψ)≠ ∅ . For a variety V of type τ we denote by V c , 2 the variety of type τ defined by all identities φ ≉ ψ from Id(V) which are either clone compatible...

Subdirect products of certain varieties of unary algebras

Miroslav Ćirić, Tatjana Petković, Stojan Bogdanović (2007)

Czechoslovak Mathematical Journal

J. Płonka in [12] noted that one could expect that the regularization ℛ ( K ) of a variety K of unary algebras is a subdirect product of K and the variety D of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties K which are contained in the generalized variety T D i r of the so-called trap-directable algebras.

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